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MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes

Beatriz Barbero-Lucas, Fernando Hernando, Helena Martín-Cruz, Gary McGuire

Abstract

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in $m$ variables. When $m=1$ our codes are MDS, and when $m=2$ and our lower bound for the minimum distance is $3$ the codes are at least Hermitian Almost MDS. For an infinite family of parameters when $m=2$ we prove that our codes beat the Gilbert-Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.

MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes

Abstract

We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in variables. When our codes are MDS, and when and our lower bound for the minimum distance is the codes are at least Hermitian Almost MDS. For an infinite family of parameters when we prove that our codes beat the Gilbert-Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.
Paper Structure (13 sections, 18 theorems, 64 equations, 2 figures, 7 tables)

This paper contains 13 sections, 18 theorems, 64 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stabilizer Quantum Codes From Generalized Monomial-Cartesian Codes
  4. Self-Orthogonality Conditions
  5. Our General Construction
  6. Our Specific Construction
  7. The Dimension
  8. We obtain MDS and Hermitian Almost MDS quantum codes
  9. MDS
  10. Hermitian Almost MDS
  11. When $m=2$ we can beat Gilbert-Varshamov Bound
  12. $d=3$
  13. Examples

Key Result

Theorem 2.1

Let $C$ be a linear $[n,k,d]$ error-correcting code over the field ${\mathbb{F}}_{q^2}$ such that $C\subseteq C^{\perp_h}$. Then, there exists an $[[n,n-2k,\geq d^{\perp_h}]]_q$ stabilizer quantum code, where $d^{\perp_h}$ stands for the minimum distance of $C^{\perp_h}$.

Figures (2)

  • Figure 1: Grid representation of $E$, where $m=2$, $a_1=8$, $a_2=6$ and $\Delta=\left(\{0,1,2\}\times\{0,1\}\right) \cup \{(0,2),(1,2)\}$.
  • Figure 2: Sets $\Delta_3$, $\Delta_4$ and $\Delta_5$, where $m=2$, $a_1=8$ and $a_2=6$.

Theorems & Definitions (39)

  • Theorem 2.1: AlyKetkar
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • Corollary 2.7
  • Remark 2.8
  • ...and 29 more